Spaces of operator-valued functions measurable with respect to the strong operator topology

Abstract

Let $X$ and $Y$ be Banach spaces and $(\Omega ,\Sigma ,\mu )$ a finite measure space. In this note we introduce the space $L^p[\mu ;L(X,Y)]$ consisting of all (equivalence classes of) functions $\Phi :\Omega \mapsto L(X,Y)$ such that $\omega \mapsto \Phi (\omega )x$ is strongly $\mu$-measurable for all $x\in X$ and $\omega \mapsto \Phi (\omega )f(\omega )$ belongs to $L^1(\mu ;Y)$ for all $f\in L^{p^{\prime }}(\mu ;X)$, $1/p+1/p^{\prime }=1$. We show that functions in $L^p[\mu ;Ł(X,Y)]$ define operator-valued measures with bounded $p$-variation and use these spaces to obtain an isometric characterization of the space of all $L(X,Y)$-valued multipliers acting boundedly from $L^p(\mu ;X)$ into $L^q(\mu ;Y)$, $1\le q<p<\infty$.

Minor revisions; to appear in the proceedings of 3rd Meeting on Vector Measures, Integration and Applications (Eichstaett, 2008)